Verified decision procedures for MSO on words based on derivatives of regular expressions

Abstract

Monadic second-order logic on finite words (MSO) is a decidable yet expressive logic into which many decision problems can be encoded. Since MSO formulas correspond to regular languages, equivalence of MSO formulas can be reduced to the equivalence of some regular structures (e.g. automata). This paper presents a verified functional decision procedure for MSO formulas that is not based on automata but on regular expressions. Functional languages are ideally suited for this task: regular expressions are data types and functions on them are defined by pattern matching and recursion and are verified by structural induction.Decision procedures for regular expression equivalence have been formalized before, usually based on Brzozowski derivatives. Yet, for a straightforward embedding of MSO formulas into regular expressions an extension of regular expressions with a projection operation is required. We prove total correctness and completeness of an equivalence checker for regular expressions extended in that way. We also define a language-preserving translation of formulas into regular expressions with respect to two different semantics of MSO. Our results have been formalized and verified in the theorem prover Isabelle. Using Isabelle's code generation facility, this yields purely functional, formally verified programs that decide equivalence of MSO formulas.

Symbolic Regex Matcher

“…SRM meets all of the above expectations. Derivatives of regular expressions have been studied before in the context of matching of regular expressions, but only in the functional programming world [2,6] and in related domains [7]. Compared to earlier derivative based matching engines, the new contribution of SRM is that it supports match generation not only match detection, it supports extended features, such as bounded quantifiers, Unicode categories, and case insensitivity, it is .NET compliant, and is implemented in an imperative language.…”

Verified decision procedures for MSO on words based on derivatives of regular expressions

“…Thereby, ψ 1 , ψ 2 , and ψ 3 were processed over Σ = {a}, ϕ 1 was processed over Σ = P{P}, ϕ 2 over Σ = P{P 1 , P 2 } and finally ϕ 3 over Σ = P{P 1 , P 2 , P 3 }. The column "ICFP 2013" recapitulates the running times from the earlier unoptimized version of this procedure (Traytel & Nipkow, 2013).…”

“…A short case study of the decision procedure is given in Section 6. This paper is an extended and revised version of the homonymous ICFP 2013 functional pearl (Traytel & Nipkow, 2013). The new contributions are mostly actual changes to the decision procedure aiming to improve on both, performance and presentation:…”

LL(1) Parsing with Derivatives and Zippers

“…In 1964, Brzozowski introduced the concept of a derivatives of regular expressions [7]. This concept has proven successful in many formal proofs of parsing regular expressions and their generalisations [4,44,55,56].…”